Generally it's difficult to characterize rings isomorphic to an endomorphism ring of an abelian group. Interest in such problems was sparked by a problem given by Fuchs in his widely-read monograph *Abelian Groups*, cf. the excerpt below from the introduction to the paper [1]

The notion of an E-ring goes back to a
seminal paper of Schultz [20] written
in response to Problem 45 in the
well-known book `Abelian Groups' by
Laszlo Fuchs [11]. In this paper
Schultz distinguished between two
possibly different approaches, the
first we will continue to call an
E-ring, while the second we shall
refer to as a generalized E-ring. Thus
a ring R is said to be an E-ring if R
is isomorphic to the endomorphism ring
of its underlying additive group, R+,
via the mapping sending an element r
$\in$ R to the endomorphism given by
left multiplication by r, whilst R is
a generalized E-ring if some
isomorphism, not necessarily left
multiplication, exists between R and
its endomorphism ring End(R+). Since
right multiplication is always an
endomorphism, it is not difficult to
see that E-rings are necessarily
commutative. The existence of a
non-commutative generalized E-ring has
recently been established [15], and so
it follows that the class of
generalized E-rings is strictly larger
than the class of E-rings.

Since Schultz's original paper there
has been a great deal of interest in
E-rings and some natural
generalizations, see e.g.
[1,2,4,6,8-10,17,19,21]. A notable
feature of much of this recent work
has been the use of so-called
realization theorems, whereby a
cotorsion-free ring is realized, using
combinatorial ideas derived from
Shelah's Black Box - see e.g. [7] for
details of this technique - as the
endomorphism ring of an Abelian group.
This present work arose from an
observation of the second author in
response to a question from the first
about the existence of generalized
E-algebras over the ring $J_p$ of
p-adic integers; see [16] for further
details. A natural question which
arises, is to what extent is it
necessary for a ring to be
cotorsion-free in order to be a
generalized E-ring and the principal
objective of this work is to
characterize generalized E-rings
`modulo cotorsion-free groups.' The
characterization is quite elementary
but seems to have been overlooked
heretofore. It should be noted that
Bowshell and Schultz showed in [2]
that a reduced cotorsion E-ring has
the form $\prod_{p \in U} {\mathbb
> Z}(p^{k_p}) \oplus \prod_{p\in V} J_p$
where $U,V$ are disjoint sets of
primes.

1 R. Gobel, B. Goldsmith.

Classifying E-algebras over Dedekind domains

Jnl. Algebra, Vol. 306, 2006, 566-575